1.

What is a nonsingular matrix? Give an example of a $$2 \times 2$$ nonsingular matrix. How do you know your example is nonsingular?

2.

What is an isometry in $$\mathbb R^n\text{?}$$ Can you give an example of an isometry in $$\mathbb R^2\text{?}$$

3.

What is an orthonormal set of vectors?

4.

What is the difference between the orthogonal group and the special orthogonal group?

5.

What is a lattice?